The location of central structures in trees

doi:10.1016/0305-0548(88)90003-2

Computers & Operations Research

Volume 15, Issue 2, 1988, Pages 103-113

https://doi.org/10.1016/0305-0548(88)90003-2 Get rights and content

Abstract

A nonlinear version of the path center location problem is analyzed. We introduce a closely related problem called the subtree location problem. Variants in which the endpoints of the central structure to be located may or may not be restricted to the vertex set are also studied. We give theoretical results for the nonlinear cases when the underlying structure is a tree with edges of varying length and linear time algorithms for the unweighted vertex case. The linear time algorithms have their basis in an efficient data structures for representing trees.

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Extensive facility location models on graphs deal with the location of a special type of subgraphs such as paths, trees or cycles and can be considered as extensions of classical point location models. Variance is one of the measures applied in models in which some equality requirement is imposed. In this paper the problem of locating a minimum variance path in a tree network is addressed, and an $O (n^{2} log n)$ time algorithm is proposed.
A review of extensive facility location in networks
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A unifying location model on tree graphs based on submodularity properties
1993, Discrete Applied Mathematics
Let $F$ be the collection of nonempty subtrees of a given tree T. Each subtree is viewed as a potential facility. Let ƒ be a real objective function defined on $F$ . The facility location model we consider is to select a subtree minimizing f. This model unifies and generalizes several facility location problems discussed in the literature. We prove that the most common objective functions used in facility location theory possess the submodularity property. In particular, the ellipsoid approach provides a unified framework for polynomial solvability.
Approximating the tree and tour covers of a graph
1993, Information Processing Letters
The tree and tour cover problems on an edge-weighted graph are to compute a minimum weight tree and closed walk, respectively, whose vertices from a vertex cover. Both problems are NP-hard. In this note we give strongly polynomial time, constant factor approximation algorithms for both problems. An interesting feature of our algorithms is how they combine approximations of other problems, namely the weighted vertex cover, traveling salesman, and Steiner tree problems.
Comments on: Extensive facility location problems on networks: an updated review
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^†: Rex K. Kincaid is Assistant Professor of Mathematics at the College of William and Mary. He received his Ph.D. in operations research in the School of Industrial Engineering and his M.S. in applied mathematics both from Purdue University. His B.A. is from DePauw University. His current research interests include network location and multiobjective optimization.

^§: Thomas L. Morin is Professor of Operations Research in the School of Industrial Engineering at Purdue University. He received a B.A. from Rutgers University and a Ph.D. in operations research from Case Western Reserve Univeristy, Professor Morin is a Fulbright Scholar who is known for his work in dynamic programming. He is a principal investigator for the Office of Naval Research's University Research Initiative in Computational Combinatorics at Purdue University.

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The location of central structures in trees

Abstract

Locations on networks: a survey. Part I : The P-center and P-median problems

Mgmt Sci.

Locations on networks: a survey. Part II: Exploiting tree network structure

Mgmt Sci.

Central paths in graphs

Transport. Sci.

Some definitions of central structures in a graph

Linear algorithms for finding the Jordan center of a tree

Transport. Sci.

The optimal location of a path or tree in a tree network

Networks

A minimal length covering subtree of a tree